Two gaussian dependency

Question

Given $x_1, x_2, x_3 \sim N(0,1)$, independently and identically distributed, and $y = x_1\sin(x_3) + x_2\cos(x_3)$.

I know $Y$ is Gaussian and independent of $x_3$.

I need to check if $Y$ and $x_1$ are independent.

I see $E[yx_1] = 0$, but I don't know if they are jointly Gaussian, so I can't use that to claim independence.

Also, I think they are dependent since $E[y|x_1=0] = x_2\sin(x_3)$ and $E[y|x_1=1] = \cos(x_3) + x_2\sin(x_3)$, which is different from $E[y|x_1=0]$... so I see dependency.

Answer 1

To determine whether $Y$ and $x_1$ are independent, you need to consider whether the conditional distribution of $Y$ given $x_1$ is the same as the marginal distribution of $Y$.

Given $x_1, x_2, x_3$ are i.i.d. standard normal variables and $Y = x_1 \sin(x_3) + x_2 \cos(x_3)$, you correctly observe that $Y$ is Gaussian and independent of $x_3$, which means that $Y$ is also Gaussian.

Now, to check for independence between $Y$ and $x_1$, you need to verify if the conditional distribution of $Y$ given $x_1$ is the same as the marginal distribution of $Y$.

You've correctly noted that $E[Yx_1] = 0$, which implies that $Y$ and $x_1$ are uncorrelated. However, uncorrelatedness is not enough to establish independence.

You've also noticed that $E[Y|x_1=0]$ and $E[Y|x_1=1]$ are different, which suggests that the conditional distributions of $Y$ given different values of $x_1$ are not the same, indicating dependency.

Based on your analysis, $Y$ and $x_1$ are dependent. The conditional distribution of $Y$ given $x_1$ varies depending on the value of $x_1$, which indicates dependence between $Y$ and $x_1$.

Answer 2

Although $E(YX_1)=0$ and $Y,X_1\sim N(0,1)$, Y and $X_1$ are not independent. Actually $$E(e^{sY+tX_1})=E\left[E(e^{sY+tX_1}|X_3)\right]=e^{\frac{s^2}{2}}e^{\frac{t^2}{2}}E(e^{st \cos X_3})$$Since $E(e^{st \cos X_3})\neq 1$ this proves that $Y,X_1$ are not independent and that the pair $(Y,X_1)$ is not Gaussian. Elegant exercise.